ISSN 1936-5098

CAE Working Paper #10-02

On Equitable Social Welfare Functions Satisfying the Weak Pareto Axiom:A Complete Characterization

by

Ram Sewak Dubeyand

Tapan Mitra

July 2010

On Equitable Social Welfare Functions

satisfying the Weak Pareto Axiom: A

Complete Characterization∗

Ram Sewak Dubey†and Tapan Mitra ‡

Abstract

The paper examines the problem of aggregating infinite utilitystreams with a social welfare function which respects the Anonymityand Weak Pareto Axioms. It provides a complete characterization ofdomains (of the one period utilities) on which such an aggregationis possible. A social welfare function satisfying the Anonymity andWeak Pareto Axioms exists on precisely those domains which do notcontain any set of the order type of the set of positive and negativeintegers. The criterion is applied to decide on possibility and impos-sibility results for a variety of domains. It is also used to provide analternative formulation of the characterization result in terms of theaccumulation points of the domain.

Journal of Economic Literature Classification Numbers: D60, D70,D90.

Keywords and Phrases: Anonymity Axiom, Weak Pareto Axiom,Social Welfare Function, Infinite Utility Streams, Domain Restric-tions, Order Types, Accumulation Points.

∗We would like to thank Kaushik Basu and Kuntal Banerjee for helpful discussions.†Department of Economics, Cornell University, Ithaca, NY 14853, USA; E-mail:

rsd28@cornell.edu‡Department of Economics, Cornell University, Ithaca, NY 14853, USA; E-mail:

tm19@cornell.edu

1

1 Introduction

The possible conflict between equity and efficiency criteria in the aggregationof infinite utility streams has received considerable attention in the literatureon intertemporal welfare economics.1 In particular, formalizing the notion ofequity by the Anonymity axiom and of efficiency by some form of the Paretoaxiom, a number of results, indicating the impossibility of carrying out suchan aggregation, have been established.

In contrast, Basu and Mitra (2007b) have observed that with domainrestrictions, it is possible to have social welfare functions on infinite utilitystreams which satisfy simultaneously the Anonymity and Weak Pareto ax-ioms. The question arises as to what exactly is the nature of the domainrestrictions which allows such possibility results to emerge.

We consider the problem of defining social welfare functions on the set Xof infinite utility streams, where this set takes the form of X = Y N, with Ydenoting a non-empty subset of the reals and N the set of natural numbers.In discussing “domain restrictions” we refer to the set Y as the “domain” as ashort-hand, even though the domain of the social welfare function is actuallyX. This is because we would like to study the nature of Y that allows forpossibility results, and would like to give easily verifiable conditions on Ywhich can be checked, instead of conditions on the set X, which might beconsiderably harder to verify.

We start with a brief review of where the literature stands at the presenttime on this issue. When Y = N, Basu and Mitra (2007b, Theorem 3, p.77)show that there is a social welfare function on X = Y N which satisfies theAnonymity and Weak Pareto Axioms. In fact, one can write this functionexplicitly as the “min” function, noting that the minimum for every infiniteutility stream in X exists, when Y = N. On the other hand, if Y = [0, 1],Basu and Mitra (2007b, Theorem 4, p.78) also show that there is no socialwelfare function on X = Y N which satisfies the Anonymity and Weak ParetoAxioms.

This leads to the question of whether it is the countability of the set Y

1For a sample of the literature, see Diamond (1965), Koopmans (1972), Svensson(1980), Chichilnisky (1996), Lauwers (1998, 2010), Shinotsuka (1998), Basu and Mitra(2003, 2007a, 2007b), Fleurbaey and Michel (2003), Asheim and Tungodden (2004), Baner-jee (2006), Sakai (2006), Asheim, Mitra and Tungodden (2007), Bossert, Sprumont andSuzumura (2007), Zame (2007) Hara, Shinotsuka, Suzumura and Xu (2008) and Crespo,Núñez and Rincón-Zapatero (2009).

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that is crucial in allowing possibility results to emerge. This turns out tobe not the case: we show that when Y = I, where I is the set of positiveand negative integers, there is no social welfare function on X = Y N whichsatisfies the Anonymity and Weak Pareto Axioms.

In fact, we go further and provide a complete characterization of thedomains, Y, for which there exists a social welfare function satisfying theAnonymity and Weak Pareto Axioms. These are precisely those domainswhich do not contain any set of the order type of the set of positive andnegative integers.2

The characterization result provides a new perspective on known resultsin the literature as well as new results for domains for which the existingliterature has little to offer.

The possibility part of the result is especially useful since the social welfarefunction can be written in explicit form by a formula, involving a weightedaverage of the sup and inf functions on X (using simple monotone transfor-mations of the elements of X, if needed).

The characterization is given in terms of order types. This criterion is seento be applicable to decide on possibility and impossibility results for a varietyof domains. This is demonstrated by presenting a number of illustrativeexamples.

The criterion is also used to provide a reformulation of the characteri-zation result in terms of right and left accumulation points of the domain.This alternative characterization is shown to be easier to apply to domainsto decide on possibility and impossibility results.

2 Formal Setting and Main Result

2.1 Weak Pareto and Anonymity Axioms

Let R be the set of real numbers, N the set of positive integers, and I theset of positive and negative integers. Suppose Y ⊂ R is the set of all possibleutilities that any generation can achieve. Then X = Y N is the set of allpossible utility streams. If 〈xn〉 ∈ X, then 〈xn〉 = (x1, x2, ...), where, forall n ∈ N, xn ∈ Y represents the amount of utility that the generation ofperiod n earns. For all y, z ∈ X, we write y ≥ z if yn ≥ zn, for all n ∈ N; wewrite y > z if y ≥ z and y �= z; and we write y >> z if yn > zn for all n∈ N.

2The term “order type” is explained in Section 2.

3

If Y has only one element, then X is a singleton, and the problem ofranking or evaluating infinite utility streams is trivial. Thus, without furthermention, the set Y will always be assumed to have at least two distinctelements.

A social welfare function (SWF) is a mapping W : X → R. Consider nowthe axioms that we may want the SWF to satisfy. The first axiom is theWeak Pareto condition; this is a version of the Pareto axiom that has beenwidely used in the literature (see Arrow, 1963; Sen, 1977), and is probablyeven more compelling than the standard Pareto axiom.3

Weak Pareto Axiom: For all x, y ∈ X, if x >> y, thenW (x) > W (y).The next axiom is the one that captures the notion of ‘inter-generational

equity’; we shall call it the ‘anonymity axiom’.4

Anonymity Axiom: For all x, y ∈ X, if there exist i, j ∈ N suchthat xi = yj and xj = yi, and for every k ∈ N ∼ {i, j}, xk = yk, thenW (x) = W (y). 5

2.2 Domain Types

In this subsection, we recall a few concepts from the mathematical literaturedealing with types of spaces, which are strictly ordered by a binary relation.

We will say that the set A is strictly ordered by a binary relation R if R isconnected (if a, a′ ∈ Y and a �= a′, then either aRa′ or a′Ra holds), transitive(if a, a′, a′′ ∈ A and aRa′ and a′Ra′′ hold, then aRa′′ holds) and irreflexive (aRa holds for no a ∈ A). In this case, the strictly ordered set will be denotedby A(R). For example, the set N is strictly ordered by the binary relation <(where < denotes the usual “less than” relation on the reals).

We will say that a strictly ordered set A′(R′) is similar to the strictlyordered set A(R) if there is a one-to-one function f mapping A onto A′, suchthat:

a1, a2 ∈ A and a1Ra2 =⇒ f(a1)R′f(a2)

3The standard Pareto axiom is:Pareto Axiom: For all x, y ∈ X, if x > y, then W (x) > W (y).We caution the reader that in some of the literature, what we are calling “Weak Pareto”

is often called “Pareto”, with the suffix “strong” added to what we have called the “Paretoaxiom”.

4In informal discussions throughout the paper, the terms “equity” and “anonymity”are used interchangeably.

5If A and B are two subsets of S, the difference B ∼ A is the set {z : z ∈ B and z /∈ A}.This notation follows Royden (1988, p.13).

4

We now specialize to strictly ordered subsets of the reals. With Y a non-empty subset of R, let us define6 some order types as follows. We will saythat the strictly ordered set Y (<) is:

(i) of order type ω if Y (<) is similar to N(<);(ii) of order type σ if Y (<) is similar to I(<);(iii) of order type µ if Y contains a non-empty subset Y ′, such that the

strictly ordered set Y ′(<) is of order type σ.The characterization of these types of strictly ordered sets is facilitated by

the concepts of a cut, a first element and a last element of a strictly orderedset.

Given a strictly ordered set Y (<), let us define a cut [Y1, Y2] of Y (<) asa partition of Y into two non-empty sets Y1 and Y2 (that is, Y1 and Y2 arenon-empty, Y1 ∪ Y2 = Y and Y1 ∩ Y2 = ∅), such that for each y1 ∈ Y1 andeach y2 ∈ Y2, we have y1 < y2.

An element y0 ∈ Y is called a first element of Y (<) if y < y0 holds forno y ∈ Y. An element y0 ∈ Y is called a last element of Y (<) if y0 < y holdsfor no y ∈ Y.

The following result can be found in Sierpinski (1965, p. 210).

Lemma 1 A strictly ordered set Y (<) is of order type σ if and only if thefollowing two conditions hold:

(i) Y has neither a first element nor a last element.(ii) For every cut [Y1, Y2] of Y, the set Y1 has a last element and the set

Y2 has a first element.

2.3 The Characterization Result

The complete characterization result of the paper can now be stated as fol-lows.

Theorem 1 Let Y be a non-empty subset of R. There exists a social welfarefunction W : X → R (where X = Y N) satisfying the Weak Pareto andAnonymity axioms if and only if Y (<) is not of order type µ.

6The name “order type ω ” appears in Sierpinski (1965). The name “order type σ ” isour own, but it is discussed and characterized in Sierpinski (1965). The name “order typeµ ” is our own, and it appears to be the crucial concept for the problem we are studying.

5

The result implies that there is a social welfare function W : X → R

satisfying the Weak Pareto and Anonymity axioms (where X = Y N), whenY = N, but that there is no such function when Y = I. Additional examplesshow that the criterion given is easy to check to decide on possibility andimpossibility results.

3 The Possibility Result

We first present the possibility part of the result in Theorem 1 for domainsY ⊂ [0, 1].This enables one to explicitly define a social welfare function, andverify that, when the domain Y is such that Y (<) is not of type µ, thefunction satisfies the Weak Pareto and Anonymity axioms.

The explicit form of the social welfare function makes this possibility re-sult potentially useful for policy purposes. In addition, the social welfarefunction has the desirable property7 that it satisfies the following “mono-tonicity condition”:

For all x, x′ ∈ X, if x > x′, then W (x) ≥ W (x′) (M)

Proposition 1 Let Y be a non-empty subset of [0, 1], and suppose that Y (<)is not of order type µ. For x = (xn)

∞

n=1 ∈ X ≡ Y N, define:

W (x) = α inf{xn}n∈N + (1− α) sup{xn}n∈N

where α ∈ (0, 1) is a parameter. Then W satisfies the Anonymity and WeakPareto axioms.

Proof. (Anonymity) For any x ∈ X, W (x) depends only on the set{xn}n∈N. This set does not change with any finite permutation of its elements.So, W (x) also does not change with any such permutation. Thus, W satisfiesthe Anonymity axiom.

(Weak Pareto) Let x, x′ ∈ X with x′ >> x. We claim that W (x′) >W (x). Clearly, by definition of W, we have W (x′) ≥ W (x) and so if the claimis false, it must be the case that:

W (x′) = W (x) (1)

7A weak version of Pareto, which requires that the “monotonicity condition” (M),together with what we have called Weak Pareto axiom, be satisfied, is quite appealing,and has been proposed and examined by Diamond (1965).

6

Since:

inf{x′

n}n∈N ≥ inf{xn}n∈N and sup{x′

n}n∈N ≥ sup{xn}n∈N (2)

and α ∈ (0, 1), it follows that:

a ≡ inf{x′

n}n∈N = inf{xn}n∈N and b ≡ sup{x′

n}n∈N = sup{xn}n∈N (3)

Clearly a, b ∈ [0, 1] and b ≥ a. In fact, we must have b > a, since:

b ≡ sup{x′

n}n∈N ≥ x′

1 > x1 ≥ inf{x′

n}n∈N ≡ a (4)

We now break up our analysis into the following cases:(i) {x′

n}n∈N has a minimum.(ii) {x′

n}n∈N does not have a minimum.Case (ii) is further subdivided as follows:(ii) (a) {x′

n}n∈N does not have a minimum, and {xn}n∈N has a maximum.(ii) (b) {x′

n}n∈N does not have a minimum, and {xn}n∈N does not have amaximum.

In case (i), let k ∈ N be such that x′

k = min{x′

n}n∈N. Then, we have:

a ≡ inf{x′

n}n∈N = min{x′

n}n∈N = x′

k > xk ≥ inf{xn}n∈N ≡ a

a contradiction.In case (ii) (a), let s ∈ N be such that xs = max{xn}n∈N. Then, we have:

b ≡ sup{xn}n∈N = max{xn}n∈N = xs < x′

s ≤ sup{x′

n}n∈N = b

a contradiction.Finally, we turn to case (ii) (b). Choose c ∈ (a, b). Then, we can find

c < xn1< xn2

< xn3< · · · with xnk

∈ (c, b) for k = 1, 2, 3, ..., and xnk↑ b as

k ↑ ∞. Similarly, we can find x′

m1> x′

m2> x′

m3> · · · with x′

mr∈ (a, c) for

r = 1, 2, 3, ..., and x′

mr↓ a as r ↑ ∞. That is, we have:

a < · · ·x′

m3< x′

m2< x′

m1< c < xn1

< xn2< xn3

< · · · < b (5)

Consider the set Y ′ = {xn1, xn2

, xn3, ...}∪{x′

m1, x′

m2, x′

m3, ...}. Clearly, Y ′ is a

subset of Y and because of (5), we note that (A) Y ′ has neither a maximumnor a minimum, and (B) for every cut [Y ′

1 , Y′

2 ] of Y′, the set Y ′

1 has a lastelement and the set Y ′

2 has a first element. Thus, by Lemma 1, Y ′(<) is oforder type σ. This means Y (<) is of order type µ, a contradiction.

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Since we are led to a contradiction in cases (i), (ii)(a) and (ii)(b), andthese exhaust all logical possibilities, (1) cannot hold, and our claim thatW (x′) > W (x) is established.

While the possibility result in Proposition 1 is stated for domains Y ⊂[0, 1], we will see that the result actually holds for all non-empty domains Y ⊂R (as claimed in Theorem 1) because of an invariance result. This states thatany possibility result is invariant with respect to monotone transformationsof the domain.

Proposition 2 Let Y be a non-empty subset of R,X ≡ Y N, and W : X → R

be a function satisfying the Weak Pareto and Anonymity axioms. Suppose fis a monotone (increasing or decreasing) function from I to Y, where I is anon-empty subset of R. Then, there is a function V : J → R satisfying theWeak Pareto and Anonymity axioms, where J = IN.

Proof. We treat two cases: (i) f is increasing, and (ii) f is decreasing.(i) Let f be an increasing function. Define V : J → R by:

V (z1, z2, ...) = W (f(z1), f(z2), ...) (6)

Then, V is well-defined, since f maps I into Y.To check that V satisfies the Anonymity axiom, let z, z′ ∈ J, with z′r =

zs, z′

s = zr, and z′i = zi for all i �= r, s. Without loss of generality, assumer < s. Then,

V (z′1, z′

2, ...) = W (f(z′1), f(z′

2), ..., f(z′

r), ..., f(z′

s), ...)

= W (f(z′1), f(z′

2), ..., f(z′

s), ..., f(z′

r), ...)

= W (f(z1), f(z2), ..., f(zr), ..., f(zs), ...)

= V (z1, z2, ...) (7)

the second line of (7) following from the fact that W satisfies the Anonymityaxiom on X. Note that the fact that f is increasing is nowhere used in thisdemonstration.

To check that V satisfies the Weak Pareto axiom, let z, z′ ∈ J withz′ >> z. We have:

f(z′i) = f(zi) + [f(z′i)− f(zi)] = f(zi) + εi for each i ∈ N (8)

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where εi ≡ [f(z′i) − f(zi)] > 0 for each i ∈ N, since f is increasing. Conse-quently,

V (z′1, z′

2, ...) = W (f(z′1), f(z′

2), ...)

= W (f(z1) + ε1, f(z2) + ε2, ...)

> W (f(z1), f(z2), ...)

= V (z1, z2, ...) (9)

where the third line of (9) follows from the facts that W satisfies the WeakPareto axiom on X, f(zi) ∈ Y, f(zi) + εi ≡ f(z′i) ∈ Y for all i ∈ N, andεi > 0 for all i ∈ N.

(ii) Let f be a decreasing function. Define V : J → R by:

V (z1, z2, ...) = −W (f(z1), f(z2), ...) (10)

Then, V is well-defined, since f maps I into Y.One can check that V satisfies the Anonymity axiom by following the

steps used in (i) above. To check that V satisfies the Weak Pareto axiom,let z, z′ ∈ J with z′ >> z. We have:

f(z′i) = f(zi) + [f(z′i)− f(zi)] = f(zi)− εi for each i ∈ N (11)

where εi ≡ [f(zi) − f(z′i)] > 0 for each i ∈ N, since f is decreasing. Conse-quently,

−V (z′1, z′

2, ...) = W (f(z′1), f(z′

2), ...)

= W (f(z1)− ε1, f(z2)− ε2, ...)

< W (f(z1), f(z2), ...)

= −V (z1, z2, ...) (12)

where the third line of (12) follows from the facts that W satisfies the WeakPareto axiom on X, f(zi) ∈ Y, f(zi)−εi ≡ f(z′i) ∈ Y for all i ∈ N, and εi > 0for all i ∈ N. Thus, we have:

V (z′1, z′

2, ...) > V (z1, z2, ...) (13)

We can now state the possibility result claimed in Theorem 1 as follows.

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Proposition 3 Let Y be a non-empty subset of R. There exists a socialwelfare function W : X → R satisfying the Weak Pareto and Anonymityaxioms (where X ≡ Y N) if Y (<) is not of order type µ.

Proof. Let us define a function f : R → R by:

f(y) = (1

2)

[1 +

y

1 + |y|

]for all y ∈ R (14)

Clearly f is an increasing function from R to (0, 1).Denote f(Y ) by A; then A is a non-empty subset of (0, 1). We claim

that A(<) is not of order type µ. For if A(<) is of order type µ, then thereis a non-empty subset A′ of A such that A′(<) is of order type σ. DefineC = {y ∈ Y : f(y) ∈ A′}. Then C is a non-empty subset of Y and f is anincreasing function from C onto A′. Thus, C(<) is similar to A′(<) and soC(<) is of order type σ. Clearly C is a non-empty subset of Y, and so Y (<)must be of order type µ, a contradiction. This establishes our claim.

Since A is a non-empty subset of (0, 1), and A(<) is not of order type µ,we can apply Proposition 1 to obtain a function U : B → R satisfying theWeak Pareto and Anonymity axioms, where B ≡ AN.

Since f is an increasing function from Y to A and Y is a non-emptysubset of R, we can now apply Proposition 2 to obtain a functionW : X → R

satisfying the Weak Pareto and Anonymity axioms, where X = Y N.We now discuss examples which illustrate the usefulness of Proposition

3.Example 3.1:Let Y = N and X = Y N. We claim that Y (<) is not of order type µ. For

if Y (<) is of order type µ, then Y contains a non-empty subset Y ′ such thatY ′(<) is of order type σ. Thus, by Lemma 1, Y ′(<) has no first element. But,any non-empty subset of N(<) has a first element (Munkres (1975, Theorem4.1, p.32)). This contradiction establishes the claim.

Using Proposition 3, there is a function W : X → R satisfying the WeakPareto and Anonymity axioms, where X = Y N. This provides an alternativeapproach to the possibility result noted in Basu and Mitra (2007b, Theorem3, p.77), and in Lauwers (2010, p.37).

Example 3.2:Let Y be defined by:

Y = {1/n}n∈N

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and let X = Y N. We claim that Y (<) is not of order type µ. For if Y (<) isof order type µ, then Y contains a non-empty subset Y ′ such that Y ′(<) isof order type σ. Then, defining:

Z = {(1/y) : y ∈ Y ′}

we see that Z is a non-empty subset of N. Thus, Z(<) has a first elementand so Y ′(<) has a last element. But, by Lemma 1, Y ′(<) cannot have alast element. This contradiction establishes the claim.

Using Proposition 3, there is a function W : X → R satisfying the WeakPareto and Anonymity axioms, where X = Y N. This result is mentionedwithout proof in Basu and Mitra (2007b, footnote 9, p.83).

Example 3.3:Define A = {−1/n}n∈N, B = {1/n}n∈N and Y = A ∪ B,X = Y N. We

claim that Y (<) is not of order type µ. For if Y (<) is of order type µ, thenY contains a non-empty subset Y ′ such that Y ′(<) is of order type σ.

Define A′ = A ∩ Y ′ and B′ = B ∩ Y ′. If B′ is non-empty, then B′ is anon-empty subset of B, and therefore B′(<) has a last element (see Example2 above), call it b. If A′ is empty, then Y ′ = B′ and Y ′(<) has a last element,contradicting Lemma 1. If A′ is non-empty, then for every y ∈ A′, we havey < b. Thus, b is a last element of Y ′(<), contradicting Lemma 1 again.

If B′ is empty, then Y ′ = A′. Further, A′ is a non-empty subset of A, andtherefore has a first element (see Example 2 above). Thus, Y ′(<) must havea first element, contradicting Lemma 1.

The above cases exhaust all logical possibilities, and therefore our claimis established. Using Proposition 3, there is a function W : X → R satisfyingthe Weak Pareto and Anonymity axioms, where X = Y N.

4 The Impossibility Result

We will first present the impossibility part of the result in Theorem 1 forthe domain Y = I, the set of positive and negative integers. Clearly I(<) is of type σ and therefore of type µ. This enables us to illustrate ourapproach to the impossibility result in the most transparent way. We willthen use Proposition 2 to show that when an arbitrary non-empty subset, Y,of the reals is such that Y (<) is of type µ, there is no social welfare functionsatisfying the Weak Pareto and Anonymity axioms.

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Proposition 4 Let Y = I. Then there is no social welfare function W :X → R satisfying the Weak Pareto and Anonymity axioms (where X = Y N).

Proof. Suppose on the contrary that there is a social welfare functionW : X → R satisfying the Weak Pareto and Anonymity axioms (whereX ≡ Y N = I

N).Let Q be a fixed enumeration of the rationals in (0, 1). Then, we can

write:Q = {q1, q2, q3, ....}

For any real number t ∈ (0, 1), there are infinitely many rational numbersfrom Q in (0, t) and in [t, 1).

For each real number t ∈ (0, 1), we can then define the set M(t) = {n ∈N : qn ∈ (0, t)} and the sequence 〈ms(t)〉 as follows:

m1(t) = min{n ∈ N : qn ∈ (0, t)}

and for s ∈ N, s > 1,

ms(t) = min{n ∈ N ∼ {m1(t), ...,ms−1(t)} : qn ∈ (0, t)}

The sequence 〈ms(t)〉 is then well-defined, and:

m1(t) < m2(t) < m3(t)...

and M(t) = {m1(t),m2(t), ...}.For each real number t ∈ (0, 1), we can define the set P (t) = {n ∈ N :

qn ∈ [t, 1)} and the sequence 〈pr(t)〉 as follows:

p1(t) = min{n ∈ N : qn ∈ [t, 1)}

and for r ∈ N, r > 1,

pr(t) = min{n ∈ N ∼ {p1(t), ..., pr−1(t)} : qn ∈ [t, 1)}

The sequence 〈pr(t)〉 is then well-defined, and:

p1(t) < p2(t) < p3(t)...

and P (t) = {p1(t), p2(t), ...}.In order to make the exposition transparent, we now break up the proof

into four steps.

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Step 1 (Defining the sequence 〈x(t)〉)For each real number t ∈ (0, 1), we note that M(t) ∩ P (t) = ∅, and

M(t) ∪ P (t) = N. Then, we can define a sequence 〈x(t)〉 as follows:

xn(t) =

{2s− 1 if n = ms for some s ∈ N

−2r − 1 if n = pr for some r ∈ N(15)

Note that the sequence 〈xn(t)〉 will contain, by (15), all the positive oddnumbers in increasing order of magnitude with n, and all the negative oddnumbers less than (−1) in decreasing order of magnitude with n.

Step 2 (Comparing 〈x(α)〉 with 〈x(β)〉 )Let α, β be arbitrary real numbers in (0, 1), with α < β. Note that if

n ∈ M(α), then n ∈ M(β), and if n ∈ P (β) then n ∈ P (α). Since thereare an infinite number of rationals from Q in [α, β), there will be an infinitenumber of distinct elements of N in:

L(α, β) = M(β) ∩ P (α) = {n ∈ N : qn ∈ [α, β)}

For any n ∈ L(α, β), we have n ∈ M(β) but n /∈ M(α). That is, by (15),for each n ∈ L(α, β) it must be the case that xn(α) < 0 but xn(β) > 0.Consequently, one has:

xn(β) ≥ xn(α) for all n ∈ N (16)

Informally, these observations may be expressed as follows. In compar-ing the sequence 〈x(α)〉 with 〈x(β)〉, whenever 〈x(α)〉 has a positive entryfor some co-ordinate, there must be a positive entry for that co-ordinate in〈x(β)〉. There will be an infinite number of co-ordinates (switches) for which〈x(α)〉 will have a negative entry, but for which 〈x(β)〉 will have a positiveentry. For the remaining co-ordinates, both 〈x(α)〉 and 〈x(β)〉 will have nega-tive entries. Because of the switches, 〈x(β)〉 uses up the sub-indices in M(β)earlier and postpones using the sub-indices in P (β) till later compared to〈x(α)〉, leading to (16).

One can strengthen the conclusion in (16) as follows. This also formalizesthe informal observations given above. Define:

N = min{n ∈ N : n ∈ L(α, β)}

Then, by (15), we have xN(α) < 0, xN(β) > 0, and:

xN(β)− xN(α) ≥ 2 (17)

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Consider any n ∈ N with n > N. We have either (i) n ∈ M(α) or (ii)n ∈ P (α). Case (ii) can be subdivided as follows: (a) n ∈ P (α) and n ∈ P (β),(b) n ∈ P (α) and n /∈ P (β).

In case (i), we have n ∈ M(α) and so n ∈ M(β). But since an additionalelement of M(β) has been used up for index N, compared with M(α), ifn = mk(α), we must have n = mk+j(β) for some j ∈ N. Thus, by (15), wemust have:

xn(β)− xn(α) ≥ 2 (18)

In case (ii)(a), we have n ∈ P (α) and n ∈ P (β). But since an additionalelement of P (α) has been used up for index N, compared with P (β), ifn = pr(α), we must have n = pr−j(β) for some j ∈ N. Thus, by (15), wemust have:

xn(β)− xn(α) ≥ 2 (19)

In case (ii)(b), n ∈ P (α) and n /∈ P (β), so that n ∈ M(β). That is,n ∈ L(α, β). Thus, by (15), we have xn(α) < 0, xn(β) > 0, and:

xn(β)− xn(α) ≥ 2 (20)

To summarize, for all n ≥ N, we have:

xn(β)− xn(α) ≥ 2 (21)

For n ∈ N with n < N (if any), we have:

xn(β) = xn(α) (22)

Step 3 (Comparing x(α) with a finite permutation of x(β))Based on (21) and (22), we cannot say that W ( 〈xn(α)〉) < W ( 〈xn(β)〉),

by invoking the Weak Pareto Axiom, except if N = 1, where (by (21)):

xn(β)− xn(α) ≥ 2 for all n ∈ N (23)

We consider now the case in which N > 1. We will show that (21) and(22) can be used to obtain:

x′

n(β)− xn(α) ≥ 2 for all n ∈ N

where 〈x′(β)〉 is a certain finite permutation of 〈x(β)〉.Let n1, ..., nN−1 be the (N − 1) smallest elements of N (with n1 < · · · <

nN−1) for which xni(α) < 0 and xni

(β) > 0 for i ∈ {1, ..., N − 1}. Note that

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N = n1. Then, define 〈x′

n(β)〉 to be the sequence obtained by interchangingthe i th entry of 〈xn(β)〉 with the ni th entry of 〈xn(β)〉 for i = 1, ..., N − 1,and leaving all other entries unchanged.

If i ∈ {1, ..., N − 1}, and xi(α) < 0, then:

x′

i(β) = xni(β) > 0 ≥ xi(α) + 2

and if xi(α) > 0, then by (16),

x′

i(β) = xni(β) ≥ xi(β) + 2 ≥ xi(α) + 2

That is, in either case,

x′

i(β) ≥ xi(α) + 2 for all i ∈ {1, ..., N − 1} (24)

If i ∈ {1, ..., N − 1}, and xi(β) > 0, then since xni(α) < 0, we have:

x′

ni(β) = xi(β) > 0 ≥ xni

(α) + 2

Also, if xi(β) < 0, then by (22),

x′

ni(β) = xi(β) = xi(α) ≥ xni

(α) + 2

using the fact that xni(α) < 0 and i < ni.That is, in either case,

x′

ni(β) ≥ xni

(α) + 2 for all i ∈ {1, ..., N − 1} (25)

Finally, for n ∈ N with n /∈ {1, ..., N − 1} ∪ {n1, ..., nN−1}, we havex′

n(β) = xn(β), and so by (21),

x′

n(β) ≥ xn(α) + 2

Thus, we have established that:

x′

n(β) ≥ xn(α) + 2 > xn(α) + 1 for all n ∈ N

Using the Anonymity and Weak Pareto Axioms, we have:

W (〈xn(β)〉) = W (〈x′

n(β)〉) > W (〈xn(α) + 1〉) (26)

Step 4 (Non-overlapping intervals for distinct real numbers in (0,1))

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Define, for each t ∈ (0, 1), a sequence 〈zn(t)〉 by:

zn(t) = xn(t) + 1 for all n ∈ N (27)

Note that the sequence 〈zn(t)〉 is in X, and by the Weak Pareto axiom:

W (〈zn(t)〉) > W (〈xn(t)〉)

Thus, for each t ∈ (0, 1),

I(t) = [W (〈xn(t)〉),W (〈zn(t)〉)] (28)

is a non-degenerate closed interval in R.Let α, β be arbitrary real numbers in (0, 1), with α < β. Then, by (26),

W (〈xn(β)〉) > W (〈zn(α)〉) (29)

Thus, the interval I(β) lies entirely to the right of the interval I(α) on thereal line.

That is, for arbitrary real numbers α, β in (0, 1), with α �= β, the inter-vals I(α) and I(β) are disjoint. Thus, we have a one-to-one correspondencebetween the real numbers in (0, 1) (which is an uncountable set) and a setof non-degenerate, pairwise disjoint closed intervals of the real line (which iscountable). This contradiction establishes the Proposition.

We can now state the impossibility result for general domains of ordertype µ.

Proposition 5 Let Y be a non-empty subset of R such that Y (<) is of ordertype µ. Then there is no social welfare function W : X → R satisfying theWeak Pareto and Anonymity axioms (where X = Y N).

Proof. Suppose on the contrary that there is a social welfare functionW : X → R satisfying the Weak Pareto and Anonymity axioms (whereX = Y N). Since Y (<) is of order type µ, Y contains a non-empty subset Y ′

such that Y ′(<) is of order type σ. That is, there is a one-to-one mapping,g, from I onto Y ′ such that:

a1, a2 ∈ I and a1 < a2 =⇒ g(a1) < g(a2)

Thus, g is an increasing function from I to Y. Using Proposition 2, there isa function V : J → R satisfying the Weak Pareto and Anonymity axioms,

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where J = IN. But this contradicts the result proved in Proposition 4, and

establishes the result.We now discuss examples which illustrate the usefulness of Proposition

5.Example 4.1:Let Y = A ∪ B, where A = {−n/(1 + n)}n∈N and B = {n/(1 + n)}n∈N

and let X = Y N. Define f : I → R by:

f(y) =y

1 + |y|for all y ∈ I

where I ≡ {n}n∈N ∪ {−n}n∈N. Then, f is an increasing function from I

onto Y. Thus, Y (<) is similar to I(<) and is therefore of order type σ. ByProposition 5, there is no function W : X → R satisfying the Anonymityand Weak Pareto axioms.

Example 4.2:Let Y be the set of rationals in R, and let X = Y N. Then, since I ≡

{n}n∈N ∪ {−n}n∈N is a subset of Y, and I(<) is of order type σ, Y (<) isof order type µ. Thus, by Proposition 5, there is no function W : X → R

satisfying the Anonymity and Weak Pareto axioms.Example 4.3:Let Y be the set of positive rationals in R, and let X = Y N. Define

Y ′ = {1/n}n∈N ∪ {n}n∈N, and f : I → R by:

f(y) =

{y if y ∈ B

1/|y| if y ∈ A

where I ≡ A∪B, with A = {−n}n∈N, and B = {n}n∈N. Then, f is an increas-ing function from I onto Y ′. Thus, Y ′(<) is similar to I(<) and is thereforeof order type σ. Since Y ′ ⊂ Y, Y (<) is of order type µ. By Proposition 5,there is no function W : X → R satisfying the Anonymity and Weak Paretoaxioms.

Example 4.4:Let Y be the closed interval [0, 1] in R, and let and X = Y N. Define

Z = {1/n}n∈N∪{n}n∈N, Y′ to be the set of rationals in (0, 1), and f : Z → R

by:

f(y) =y

1 + yfor all y ∈ Z

Then, f is an increasing function from Z into Y ′. Thus, f(Z)(<) is similar toZ(<), which is of type σ (by Example 4.3). Since Y ′ contains f(Z), Y ′(<) isof type µ. Since Y contains Y ′, Y contains f(Z), and so Y (<) is of type µ.

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By Proposition 5, there is no functionW : X → R satisfying the AnonymityandWeak Pareto axioms. Our discussion of Example 4.4 provides an alterna-tive proof for the impossibility theorem of Basu and Mitra (2007b, Theorem4, p.78).

5 A Reformulation of the Main Result

We have demonstrated that the complete characterization result in Theorem1 can be applied to provide possibility and impossibility results for a varietyof domains. Nevertheless, it will not have escaped the reader’s attentionthat checking the criterion involves checking all possible subsets of Y anddetermining whether any of these subsets is of the order type σ, the ordertype of the set of positive and negative integers. Checking whether a set inR is of order type σ is relatively easy, given Lemma 1, but checking this forall possible subsets of Y may not be.

With this in mind, we devote this final section to a reformulation of themain result in terms of a criterion which involves looking at the accumulationpoints of Y.

For what follows, Y will be taken to be a non-empty subset of [0, 1]. Iffor an application, one encounters a non-empty subset Y of R which is nota subset of [0, 1], one can always make a change of variable in the domain(through a monotone increasing function) so that the new domain Y ′ is anon-empty subset of [0, 1]. We have, in fact, done this already in discussingexamples in Sections 3 and 4.

Because Y ⊂ [0, 1] is a subset of R, it is possible to define right accumu-lation points and left accumulation points in the same spirit as right handlimits and left hand limits.

We will say that z ∈ R is a right accumulation point of Y if given anyδ > 0, there is y ∈ Y such that:

0 < y − z < δ

Similarly, we will say that z ∈ R is a left accumulation point of Y if givenany δ > 0, there is y ∈ Y such that:

0 < z − y < δ

Denote by R the set of right accumulation points of Y and by L the set of leftaccumulation points of Y. If Y has an infinite number of elements, then (being

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bounded) it will have an accumulation point.8 Any accumulation point ofY will be either a right accumulation point or a left accumulation point orboth. Further a right or left accumulation point of Y is also an accumulationpoint of Y.

Let us denote:ρ ≡ inf R and λ ≡ supL

with the convention that if R is empty, then ρ = ∞, and if L is empty thenλ = −∞.

We can now state our characterization result as follows.

Theorem 2 Let Y be a non-empty subset of [0, 1]. There exists a socialwelfare function W : X → R (where X = Y N) satisfying the Weak Paretoand Anonymity axioms if and only if:

ρ ≡ inf R ≥ supL ≡ λ (30)

Proof. (Necessity) Suppose condition (30) is violated; that is:

inf R < supL (31)

Given the convention adopted, this means that ρ, λ are in R and ρ < λ. Itfollows that there is a right accumulation point ρ′ of Y and a left accumulationpoint λ′ of Y such that ρ′ < λ′.

Choose c ∈ (ρ′, λ′). Then, we can find c < y1 < y2 < y3 < · · · with yk ∈ Yfor k ∈ N such that yk ↑ λ′ as k ↑ ∞ (since λ′ is a left accumulation point ofY ). Similarly, we can find c > y′1 > y′2 > y′3 > · · · with y′r ∈ Y for r ∈ N suchthat y′r ↓ ρ′ as r ↑ ∞ (since ρ′ is a right accumulation point of Y ). That is,we have:

ρ′ < · · · y′3 < y′2 < y′1 < c < y1 < y2 < y3 < · · · < λ′ (32)

Consider the set Y ′ = {y1 < y2 < y3 < · · · } ∪ {y′1 < y′2 < y′3 < · · · }. Clearly,Y ′ is a subset of Y and because of (32), we note that (A) Y ′ has neither amaximum nor a minimum, and (B) for every cut [Y ′

1 , Y′

2 ] of Y′, the set Y ′

1 hasa last element and the set Y ′

2 has a first element. Thus, by Lemma 1, Y ′(<)is of order type σ. This means Y (<) is of order type µ, and by Theorem 1,there is no social welfare function W : X → R (where X = Y N) satisfyingthe Weak Pareto and Anonymity axioms.

8For the standard definition of an accumulation point of a set, see Royden (1988, p.46).

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(Sufficiency) Suppose (30) holds. We claim that Y (<) is not of ordertype µ. For if it is of order type µ, there is a non-empty subset Y ′ ⊂ Y, suchthat Y ′(<) is of order type σ. Since Y ′ ⊂ [0, 1], it has a greatest lower bound,a, and a least upper bound, b. Clearly a ≤ b.

Since Y ′(<) is of order type σ, it does not have a maximum. So, b cannotbe in Y ′. Since b is a least upper bound of Y ′, we can find y1 < y2 < y3 < · · ·with yk ∈ Y ′ for k ∈ N such that yk ↑ b as k ↑ ∞. Then, b ∈ L, and so b ≤ λ.

Since Y ′(<) is of order type σ, it does not have a minimum. So, a cannotbe in Y ′. Since a is a greatest lower bound of Y ′, we can find y′1 > y′2 > y′3 >· · · with y′r ∈ Y ′ for r ∈ N such that y′r ↓ a as r ↑ ∞. Then, a ∈ R, and soa ≥ ρ.

Thus, we have:a ≥ ρ ≥ λ ≥ b ≥ a

so that a = b. But then Y ′ must be a singleton, and therefore Y ′(<) cannotbe of order type σ. This contradiction establishes our claim. Now, applyingTheorem 1, there exists a social welfare function W : X → R (where X =Y N) satisfying the Weak Pareto and Anonymity axioms.

Remarks:We can now re-examine the examples in Sections 3 and 4 to see the

applicability of Theorem 2 in deciding on possibility and impossibility results.In the examples in Section 3, one can check that ρ ≥ λ, so by Theorem 2

there exists a social welfare function W : X → R (where X = Y N) satisfyingthe Weak Pareto and Anonymity axioms.

In Example 3.1, Y = N which is similar to Y ′ = {n/(1 + n)}n∈N . So itis enough to examine Y ′ which is a subset of [0, 1]. There is one left accumu-lation point (namely 1) and no right accumulation point. So ρ = ∞ whileλ = 1, yielding ρ ≥ λ.

In Example 3.2, Y = {1/n}n∈N , so there is one right accumulation point(namely 0) and no left accumulation point. Thus ρ = 0 while λ = −∞,yielding ρ ≥ λ.

In Example 3.3, Y = {1/n}n∈N ∪ {−1/n}n∈N , so there is one right accu-mulation point (namely 0) and one left accumulation point (namely 0). Thusρ = 0 = λ.

In the examples in Section 4, one can check that ρ < λ (for a non-emptyset Y ′ similar to Y ) so by Theorem 2 there does not exist any social welfarefunction W : X → R (where X = Y N) satisfying the Weak Pareto andAnonymity axioms.

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In Example 4.1, Y = A ∪ B, where A = {−n/(1 + n)}n∈N and B ={n/(1 + n)}n∈N . Then Y is similar to Y ′ = f(Y ), where f is given by:

f(y) = (1

2)(1 + y) for all y ∈ Y

Then Y ′ is a non-empty subset of [0, 1]. It has a right accumulation point at0 and a left accumulation point at 1. Thus ρ = 0 < 1 = λ.

In Example 4.2, Y is the set of rationals in R. Then Y is similar toY ′ = f(Y ), where f is given by (14). Then, Y ′ is a non-empty subset of[0, 1], coinciding with set of rationals in (0, 1). Thus, every point in [0, 1) isa right accumulation point, and every point in (0, 1] is a left accumulationpoint of Y ′. Thus, ρ = 0 < 1 = λ.

In Example 4.3, Y is the set of positive rationals in R. Then, Y is similarto Y ′ = f(Y ), where f is given by:

f(y) =y

1 + yfor all y ∈ Y

Then, Y ′ coincides with set of rationals in (0, 1), and (as in Example 4.2),we have ρ < λ.

In Example 4.4, Y = [0, 1]. Then, every point in [0, 1) is a right accumu-lation point, and every point in (0, 1] is a left accumulation point of Y. Thus,ρ = 0 < 1 = λ.

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